Unimodular covers of 3-dimensional parallelepipeds and Cayley sums

Abstract

We show that the following classes of lattice polytopes have unimodular covers, in dimension three: the class of parallelepipeds, the class of centrally symmetric polytopes, and the class of Cayley sums Cay(P, Q) where the normal fan of Q refines that of P . This improves results of Beck et al. (2018) and Haase et al. (2008) where the last two classes were shown to be IDP.

The complete classification of empty lattice 4-simplices

“…A first example of this is Pick's formula relating the area and number of lattice points in a lattice polygon [BR15], which can easily be derived from Euler's formula and the fact that every empty triangle is unimodular. See [KS03,SZ13,CS19,BS16] for examples where the classification of 3-dimensional empty simplices is applied to derive general results on lattice 3-polytopes.…”

Normal polytopes and ellipsoids

“…But progress in this direction is scarce. Recent works [3,10] show that all lattice 3-dimensional parallelepipeds and centrally symmetric 3-polytopes with unimodular corners have unimodular cover.…”